Definitions

Standard definition

Given a differentiable manifoldXX, or even a generalized smooth spaceXX for which this definition makes sense, a differential form on XX is a section of the exterior algebra of the cotangent bundle over XX; sometimes one refers to an exterior differential form to be more precise. One often requires differential forms to be smooth, or at least continuous, but we will state this explicitly when we want it. A differential pp-form on XX is a section of the ppth exterior power of the cotangent bundle; the natural numberp≥0p \geq 0 is the rank of the form. A general differential form is a pp-indexed sequence of differential pp-forms of which all but finitely many are zero; on a finite-dimensional manifold, this latter condition is automatic.

The spaceC∞Ω*(X)C^\infty\Omega^*(X) of smooth forms on XX may also be defined as the universal differential envelope of the space C∞Ω0(X)C^\infty\Omega^0(X) of smooth functions on XX (which are the same as the smooth 00-forms as defined above); more concretely:

It is generated by the smooth functions and three operations:

an associative binary operation of addition generalising the usual addition of functions,

in which ff is a smooth function and η\eta is an arbitrary smooth form. (Note that one often drops the ‘∧\wedge’ after a 00-form; thus, fη=f∧ηf \eta = f \wedge \eta. There is hardly any ambiguity if one drops the ‘∧\wedge’ entirely, but it's traditional.)

Although not directly stated, it can be proved that addition makes C∞Ω*(X)C^\infty\Omega^*(X) into an abelian group; in fact, it is a module of the commutative ring of smooth functions on XX. This is further a graded module, graded by the natural numbers, with the elements of grade pp being the pp-forms defined earlier; the space of these is C∞Ωp(X)C^\infty\Omega^p(X). If ω\omega is a pp-form and η\eta is a qq-form, we have:

dω\mathrm{d}\omega is a (p+1)(p+1)-form,

ω∧η\omega \wedge \eta is a (p+q)(p+q)-form.

The law

ddη=0\mathrm{d}\mathrm{d}\eta = 0

holds for any form η\eta, but the other laws become more complicated; if ω\omega is a pp-form and η\eta is a qq-form, then we get

That is, C∞Ω*(X)C^\infty\Omega^*(X) is a skew-commutative algebra over the ring of smooth functions, equipped with a derivationd\mathrm{d} of degree 11. In fact, the description above in terms of generators and relations makes it the free skew-commutative algebra over that ring equipped with such a derivation. (Or if it doesn't, then it's because I left something out of that description.)

More general forms (in Ω*(X)\Omega^*(X)) can be recovered as sums of terms, each of which is the wedge product of a function and a smooth form. (This can also be seen as a special case of a vector-valued form as below.) One can still define the exterior derivative of a C1C^1 (once continuously differentiable) form; in general, the differential of a CkC^k form is a Ck−1C^{k-1} form. If XX is not a smooth manifold but only CkC^k for some 1≤k<∞1 \leq k \lt \infty, then one has to take more care here, but the definition of the skew-commutative algebra of CkC^k differential forms can still be made to work.

Given local coordinates (x1,…,xn)(x^1, \ldots, x^n) on a patch UU in an nn-dimensional manifold XX, any differential form η\eta on UU can be expressed uniquely as a sum of 2n2^n terms

η=∑IηI∧dxI, \eta = \sum_I \eta_I \wedge \mathrm{d}x^I ,

where II runs over increasinglists of indices from (1,…,n)(1,\ldots,n), each ηI\eta_I is a function on UU (continuous, smooth, etc according as η\eta is), and

(for pp the length of the list II) is simply an abbreviation. For a pp-form, there are (np)\left(n \atop p\right) terms that appear.

Twisted and vector-valued forms

Recall that a differential form on XX is a section of the exterior algebra of the cotangent bundle over XX; call this bundle Λ\Lambda. Then given any vector bundleVV over XX, a VV-valued form on XX is a section of the vector bundle V⊗ΛV \otimes \Lambda. The wedge product of a VV-valued form and a V′V'-valued form is a (V⊗V′)(V \otimes V')-valued form, but if there is a commonly used multiplication map V⊗V′→WV \otimes V' \to W, then we may think of their wedge product as a WW-valued form.

Of particular importance are LL-valued forms when LL is a line bundle; these are also called LL-twisted forms. (Compare the notion of twisted form in a more general context.) In local coordinates, a twisted form looks just like an ordinary form, once you choose a nonzero vector in LL as a basis. Therefore, they can seem sneaky and confusing sometimes when you realise that they do not behave in the same way!

Let Ψ\Psi be the pseudoscalar? bundle; that is, a section of Ψ\Psi (a pseudoscalar field) is given locally by a simple scalar field (a real-valued function) for each orientation of a local patch, with opposite orientations giving oppositely-signed scalars. A pseudoform is a Ψ\Psi-twisted form.

On an nn-dimensional manifold XX, the space Ωn(X)\Omega^n(X) of nn-forms is itself a line bundle; a pp-form twisted by this line bundle is a densitised form. Sometimes an nn-form is itself called a density. Actually, as we will see under integration below, it is really an nn-pseudoform that should be called a density, but that is not the traditional terminology.

Given any real number ww, there is a line bundle called the line bundle of ww-weighted? scalars; a form twisted by this line bundle is a ww-weighted form. Note that a 00-weighted form is just an ordinary form; also, an nn-pseudoform turns out to be equivalent to a 11-weighted 00-form. (And thus a densitised form is equivalent to a 11-weighted pseudoform.)

The line bundle of nn-pseudoforms (that is of 11-weighted 00-forms) is the absolute value of the line bundle of nn-forms (that is of densitised 00-forms), so we may take the absolute value of one of either and get one of the latter. (Similarly, the line bundle of 00-forms is the absolute value of the line bundle of pseudo-00-forms; that is, the trivial bundle is the absolute value of the pseudoscalar bundle.)

Urs, do you know where the need for orientation comes in here? I don't follow it in enough detail to see, although I intend to read Moerdijk–Reyes. —Toby

Eric: I’m probably confused, but if σn\sigma_n is a morphism in Pn(X)P_n(X), then (unless XX is a directed space), the opposite σn−1\sigma_n^{-1} is also in Pn(X)P_n(X) and I think ω(σn)=−ω(σn−1)\omega(\sigma_n) = -\omega(\sigma_n^{-1}).

Eric: I think it is a good question. Maybe we should keep the query box here until the answer is incorporated in the page.

Urs Schreiber: here is my reply, that I originally posted at latest changes. I’ll try to eventually work this into the main text of the entry

The orientation of the diffential form corresponds to the inherent orientation of k-morphisms: as we identify the differential form with a smooth functor on the path n-groupoid, that path n-groupoid necessarily has orientedkk-volumes as its k-morphisms, simply because these kk-morphisms need to come with information about their (higher categorical) source and target.

To get pseudo-differential forms that may be integrated also over non-oriented and possibly non-orientable manifolds one needs to consider parallel transport functors not with coefficients in just Bnℝ\mathbf{B}^n \mathbb{R} coming from the crossed complex

(ℝ→*→⋯→*)
(\mathbb{R} \to {*} \to \cdots \to {*})

but the more refined crossed complex

(ℝ→*→⋯→ℤ2)
(\mathbb{R} \to {*} \to \cdots \to \mathbb{Z}_2)

where the ℤ2\mathbb{Z}_2-factor acts by sign reversal on ℝ\mathbb{R} (one can also use U(1)U(1) instead of ℝ\mathbb{R}, this way [Pn(−),BnU(1)][P_n(-), \mathbf{B}^n U(1)] becomes the Deligne complex and knows not just about differential forms but about U(1)U(1)n−1n-1-gerbes with connection even).

A little bit of discussion of this unoriented case is currently at orientifold. There for the case n=2n=2.

Note that an nn-morphism in Pn(X)P_n(X) is an oriented nn-dimensional submanifold of XX.

Such a functor (as described in more detail at connection on a bundle) assigns a real number to each parametrised nn-dimensional cube of XX, that is a subspace by a smooth map Σ:[0,1]n→X\Sigma\colon [0,1]^n \to X. If the differential form that this nn-functor defines is denoted ω∈Ωn(X)\omega \in \Omega^n(X), then this real number is denoted by the integral

∫[0,1]nΣ*ω.
\int_{[0,1]^n} \Sigma^* \omega
\,.

This integral in turn encodes the nn-functoriality of the nn-functor: it effectively says that * if we decompose the standard nn-cube [0,1]n[0,1]^n into NnN^n little subcubes (Ck)k∈ℕn(C_k)_{k\in \mathbb{N}^n} for N∈ℕN \in \mathbb{N} * and apply the nn-functor to each of these to obtain a result (a real number) to be denoted ω(Ck)\omega(C_k); * then by nn-functoriality the result of the application of the functor to the full Σ\Sigma is the composition of all the ω(Ck)\omega(C_k) in ℝ\mathbb{R}. i.e. their sum

Since one can let NN increase arbitrarily in this prescription – N→∞N \to \infty – it follows that the value of the functor on Σ\Sigma is already determined by all its values on all “infinitesimal nn-cubes” in some sense.

The notion of differential form is the one that makes this precise: a differential form is a rule for assigning to each “infinitesimal nn-cube” a number.

In synthetic differential geometry the statement is in essence the same one, but the difference is that there the notion of “infinitesimal nn-cube” has a concrete meaning on the same footing of other nn-cubes. If DnD^n denotes the abstract infinitesimal nn-cube in this context, then the mapping space XDnX^{D^n} of morphisms from Dn→XD^n\to X is the nn-fold tangent bundle of XX and a differential form is precisely nothing but a morphism

ω:XDn×Dn→ℝ
\omega\colon X^{D^n} \times D^n \to \mathbb{R}

(where ℝ\mathbb{R} is now the synthetic differential version of the real numbers) subject to three constraints. (These constraints can be seen as the infinitesimal analog of the nn-functoriality discussed above).

Thus, the operation that maps XX to Ω*(X)\Omega^*(X) extends to a contravariant functorΩ*\Omega^*. Perhaps confusingly, forms are traditionally known in physics as ‘covariant’ concepts, because of how the components transform under a change of coordinates. (Ultimately, this confusion goes back to that between active and passive coordinate transformations.)

Note that twisted and (more general) vector-valued forms cannot be pulled back so easily. One needs some extra structure on ff to do so; see the discussion of integration of pp-pseudoforms at integration of differential forms for an example.

If we wish to integrate untwisted (or differently twisted or vector-valued) forms and/or forms of smaller rank, then we may do so on submanifolds of XX equipped with some appropriate structure. In particular, if XX itself is equipped with an orientation, then nn-pseudoforms on XX are the same as (untwisted) nn-forms, and so we can integrate those on XX. See integration of differential forms for the general case.

Zoran Škoda: Should maybe this entry have a discussion on heuristics behind the usual trick in supersymmetry which asserts that the inner hom for supermanifolds, gives the statement that the algebra of smooth differential forms on MM is the space of functions on the odd tangent bundle ΠTM\Pi T M? I am not the most competent to do this succinctly enough…

Zoran Škoda: By no means. Ordinary differential forms on ORDINARY manifolds are the same as functions on odd tangent bundle. I did not want to say anything about the generalization of differential forms on supermanifolds. So it is NOT a different notion, but a different way to define it. If going to toposes hence synthetic framework is not separated why would be separated the equality which involves a parity trick…

Toby: Ah, I see; your MM above need not be super, and it still works. Then yes, that should be mentioned here too.

As smooth differential forms are the cochains in de Rham cohomolgy, the theory of integration of forms allows us to interpret relatively compact oriented submanifolds as chains on XX, giving us a homology theory. Combining these, we have Stokes's theorem

∫∂Rω=∫Rdω, \int_{\partial{R}} \omega = \int_R \mathrm{d}\omega ,

where ∂R\partial{R}, which may be interpreted as the boundary of RR, is also called the codifferential as it is dual to d\mathrm{d}.

Much fun discussion between Eric Forgy, Toby Bartels, and John Baez, about whether integration of forms or pseudoforms is most fundamental (and about whether twisted forms in general are useful and interesting geometric objects or the bastard spawn of hell) may be found in this giant Usenet thread. More specifically: